Showing papers in "Journal of the American Statistical Association in 1970"
TL;DR: In this paper, it is shown that the residual autocorrelations are to a close approximation representable as a singular linear transformation of the auto-correlations of the errors so that they possess a singular normal distribution.
Abstract: Many statistical models, and in particular autoregressive-moving average time series models, can be regarded as means of transforming the data to white noise, that is, to an uncorrelated sequence of errors. If the parameters are known exactly, this random sequence can be computed directly from the observations; when this calculation is made with estimates substituted for the true parameter values, the resulting sequence is referred to as the "residuals," which can be regarded as estimates of the errors. If the appropriate model has been chosen, there will be zero autocorrelation in the errors. In checking adequacy of fit it is therefore logical to study the sample autocorrelation function of the residuals. For large samples the residuals from a correctly fitted model resemble very closely the true errors of the process; however, care is needed in interpreting the serial correlations of the residuals. It is shown here that the residual autocorrelations are to a close approximation representable as a singular linear transformation of the autocorrelations of the errors so that they possess a singular normal distribution. Failing to allow for this results in a tendency to overlook evidence of lack of fit. Tests of fit and diagnostic checks are devised which take these facts into account.
2,533 citations
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2,001 citations
TL;DR: In this paper, the character of the covariance matrix is investigated and it is shown that the matrix may exhibit a more general character than is typically implied to be essential, and the necessary and sufficient condition is the equality of variances of differences for all pairs of treatment measures assumed to be correlated.
Abstract: Investigation is made of the character of the covariance matrix which will result in exact F-distributions for the treatments and interaction variance ratios in repeated measurements designs. It is shown, assuming multivariate normality, that the matrix may exhibit a more general character than is typically implied to be essential. Equality of variances and equality of covariances, with identical matrices for all levels of a second treatment factor, are sufficient but not necessary conditions. The necessary and sufficient condition is the equality of variances of differences for all pairs of treatment measures assumed to be correlated. An alternative statement is that the Box-Geisser-Greenhouse parameter e = 1.0. A test is described which bears on the tenability of this condition.
955 citations
TL;DR: In this article, it is shown that isometry is possible with respect to at most one size variable, or in other words that shape will always be related to a variety of size variables.
Abstract: Size-related shape changes in animals are studied within a general framework of size variables and shape vectors. Isometry, or independence of shape and size, is defined as the independence of some (all) shape vector(s) from a particular size variable. With mild restrictions it is shown that isometry is possible with respect to at most one size variable, or in other words that shape will always be related to a variety of size variables. The choice of a size variable is a hitherto neglected, but important, part of an allometric study. The use of functional relationships in allometry is contrasted with the approach developed here. Also, size and shape variables are used in characterizations of the lognormal, gamma and generalized gamma distributions. The results, given in a biological context, are of interest in size and shape studies generally.
701 citations
TL;DR: For the m-way contingency table, the authors discusses both the direct estimation of the multiplicative interactions among the m variables, and the indirect testing of hypotheses pertaining to these interactions, including hypotheses that can be expressed in terms of one or more of the following kinds of concepts: (a) the usual concepts of independence and equiprobability; (b) concepts describing conditional properties (e.g., conditional independence) pertaining to a subset of the m variable, given the level of some of the remaining variables; (c) concepts related to the usual logit-analysis or
Abstract: For the m-way contingency table, we discuss both the direct estimation of the multiplicative interactions among the m variables, and the indirect testing of hypotheses pertaining to these interactions. We consider, among other things, hierarchical hypotheses pertaining to the interactions among the m variables, including hypotheses that can be expressed in terms of one or more of the following kinds of concepts: (a) the usual concepts of independence and equiprobability; (b) concepts describing conditional properties (e.g., conditional independence) pertaining to a subset of the m variables, given the level of some of the remaining variables; (c) concepts related to the usual logit-analysis or to a generalized form of logit-analysis; and (d) concepts related to a more general log-linear model. Methods of partitioning these hypotheses are introduced which provide, among other things, insight into the relationship between tests applied to the m-way table and tests applied to marginal tables formed ...
696 citations
TL;DR: In this paper, the authors estimate the implicit prices of specific aspects of the bundles of residential services consumed by urban households by regressing market price of owner-and renter-occupied dwelling units on measures of qualitative and quantitative dimensions of the housing bundle.
Abstract: Based on an extensive sample of individual dwelling units, this article estimates the market value, or the implicit prices, of specific aspects of the bundles of residential services consumed by urban households. Quantitative estimates were obtained by regressing market price of owner- and renter-occupied dwelling units on measures of the qualitative and quantitative dimensions of the housing bundle. The measures of residential quality associated with individual dwelling units were obtained by using factor analysis to aggregate some 39 indexes of the quality of narrowly defined aspects of the dwelling units, structures, parcels, and micro-neighborhoods. The analysis indicates that the quality of the bundle of residential services has about as much effect on the price of housing as such objective aspects as the number of rooms, number of bathrooms, and lot size. The analysis also confirms the influence of neighborhood schools on the value of residential properties and indicates that rental propert...
665 citations
TL;DR: For example, Tufte as discussed by the authors observes that the popular media in the U.S. considers the scatterplot beyond the common reader, and there is hope that the statistical literacy movement will eventually force a well-justified reassessment.
Abstract: Plot interpretation is an interesting topic because the ability to interpret plots is a measure of scientific literacy and because plots themselves can be informative. Tufte (1983) observes that the popular media in the U.S. considers the scatterplot beyond the common reader. Noting that some elementary school curricula include Cartesian plots, and even box and whisker plots, there is hope that the statistical literacy movement will eventually force a well-justified reassessment.
476 citations
431 citations
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357 citations
TL;DR: In this paper, a new method known as MINQUE (Minimum Norm Quadratic Unbiased Estimation) is introduced for the estimation of the heteroscedastic variances.
Abstract: Let Y=Xβ+e be a Gauss-Markoff linear model such that E(e) = 0 and D(e), the dispersion matrix of the error vector, is a diagonal matrix δ whose ith diagonal element is σi 2, the variance of the ith observation yi. Some of the σi 2 may be equal. The problem is to estimate all the different variances. In this article, a new method known as MINQUE (Minimum Norm Quadratic Unbiased Estimation) is introduced for the estimation of the heteroscedastic variances. This method satisfies some intuitive properties: (i) if S 1 is the MINQUE of Σ piσi 2 and S 2 that of Σqiσi 2 then S 1+S 2 is the MINQUE of σ(pi + qi )σi 2, (ii) it is invariant under orthogonal transformation, etc. Some sufficient conditions for the estimation of all linear functions of the σi 2 are given. The use of estimated variances in problems of inference on the β parameters is briefly indicated.
TL;DR: In this paper, an asymptotic χ2 test for the equality of two correlation matrices is derived and the test statistic has the form of a standard normal theory statistic with a correction term added.
Abstract: An asymptotic χ2 test for the equality of two correlation matrices is derived. The key result is a simple representation for the inverse of the asymptotic covariance matrix of a sample correlation matrix. The test statistic has the form of a standard normal theory statistic for testing the equality of two covariance matrices with a correction term added. The applicability of asymptotic theory is demonstrated by two simulation studies and the statistic is used to test the difference in the factor patterns resulting from a set of tests given to retarded and non-retarded children. Two related tests are presented: a test for a specified correlation matrix and a test for equality of correlation matrices in two or more populations.
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TL;DR: In this article, a double sampling scheme is presented to estimate the proportion of units which belong to one of two mutually exclusive categories, and the maximum likelihood estimate of p is derived along with its asymptotic variance.
Abstract: Two measuring devices are available to classify units into one of two mutually exclusive categories. The first device is an expensive procedure which classifies units correctly; the second device is a cheaper procedure which tends to misclassify units. To estimate p, the proportion of units which belong to one of the categories, a double sampling scheme is presented. At the first stage, a sample of N units is taken and the fallible classifications are obtained; at the second stage a subsample of n units is drawn from the main sample and the true classifications are obtained. The maximum likelihood estimate of p is derived along with its asymptotic variance. Optimum values of n and N which minimize the measurement costs for a fixed variance of estimation and which minimize the precision for fixed cost are derived. This double sampling scheme is compared to the binomial sampling scheme in which only true measurements are obtained.
TL;DR: In this article, a new model is developed and its performance compared with a model proposed by Rao and Kupper, and the likelihood ratio test of the hypothesis of equal preferences is shown to have the same asymptotic efficiency as that for the Rao-Kupper model.
Abstract: This study is concerned with the extension of the Bradley-Terry model for paired comparisons to situations which allow an expression of no preference. A new model is developed and its performance compared with a model proposed by Rao and Kupper. The maximum likelihood estimates of the parameters are found using an iterative procedure which, under a weak assumption, converges monotonically to the solution of the likelihood equations. It is noted that for a balanced paired comparison experiment the ranking obtained from the maximum likelihood estimates agrees with that obtained from a scoring system which allots two points for a win, one for a tie and zero for a loss. The likelihood ratio test of the hypothesis of equal preferences is shown to have the same asymptotic efficiency as that for the Rao-Kupper model. Two examples are presented, one of which introduces a set of data for an unbalanced paired comparison experiment. Initial applications of the test of goodness of fit suggest that the propos...
TL;DR: In this article, the authors discuss the problem of comparing the means of two populations when the ratio of their variances is unknown, in terms of interval estimation and testing, and compare six solutions.
Abstract: The Behrens-Fisher problem is that of comparing the means of two populations when the ratio of their variances is unknown. Solutions are discussed in terms of interval estimation and testing. Of the six solutions compared, three are preferred by the writer and judged practical, and for these, approximations are given for the power of the tests.
TL;DR: In this paper, the authors investigated empirical evidence on the structure of price expectations in the United States during the post-Korean War period using semiannual data which describe price expectations for six months and twelve months ahead.
Abstract: This article investigates empirical evidence on the structure of price expectations in the United States during the post-Korean War period. The study utilizes semiannual data which describe price expectations for six months and twelve months ahead. The objective is to use these two sets of data to test some of the well-known expectational hypotheses. Incidental to this we determine whether expectations satisfy the rationality hypothesis, and we briefly consider the accuracy of the predictions.
TL;DR: In this paper, the results of Finney minimum variance unbiased estimators, τ, of parametric functions of the type are established, and expressions for the variance of τ and an unbiased estimator of the variance are obtained.
Abstract: Let Y = ey where the usual assumptions of the general linear models: y = Xa+u, u ∼ N(0, σ2 I) are made. Extending the results of Finney minimum variance unbiased estimators, τ, of parametric functions of the type are established. Expressions for the variance of τ and an unbiased estimator of the variance are obtained. The results are in terms of functions whose numerical values are tabulated.
TL;DR: Balanced repeated replications (BRR) is a general method for computing standard errors as discussed by the authors, which is useful when mathematical distribution theory is impractical or lacking, and especially for analytical statistics based on complex samples where clustering destroys the independence of observations.
Abstract: Balanced repeated replications (BRR) is a general method for computing standard errors. It is useful when mathematical distribution theory is impractical or lacking, and especially for analytical statistics based on complex samples where clustering destroys the independence of observations. Presented are results of methods used to measure standard errors of regression coefficients for several multivariate techniques. The basic designs of the several samples comprised two primary selections (PS) per stratum. Each replication was a half-sample, created by selecting one PS from each stratum. The variance of the coefficient , estimated from the entire sample, is measured by , where bj is the same estimator based on a half-sample. To increase the precision of the variance estimate, select k repeated replications and obtain the mean of the k computed variances, . Balanced repeated replications reduce the number of repetitions needed; e.g., 48 balanced replications sufficed for 47 strata in our samples....
TL;DR: In this paper, a locally most powerful test is developed for the hypothesis that a slope coefficient in a linear time series model is stable, against the alternative that the slope shifts exactly once somewhere in the series.
Abstract: A locally most powerful test is developed for the hypothesis that a slope coefficient in a linear time series model is stable, against the alternative that the slope shifts exactly once somewhere in the series. Analysis of the procedure using artificial data indicates good power characteristics even when the ratio of the shift size to the error variance is moderate—especially if the shift does not occur very near either end of the series. Power also depends on the pattern of the independent variables and on whether the error variance is known or must be estimated using the residuals about the regression line.
TL;DR: In this article, the problem of making statistical inferences about an unknown value of x corresponding to one or more additional observed values of y is considered, and it is shown that the inverse estimator is Bayes with respect to a particular informative prior.
Abstract: The model considered in this paper is simple linear regression , and the problem is to make statistical inferences about an unknown value of x corresponding to one or more additional observed values of y. The maximum likelihood estimator x of x and the classical (1 - α) 100% confidence set S for x have some undesirable properties. For example, has infinite mean square error and > 0. The purpose of this paper is to demonstrate that insight and understanding, as well as a useful class of solutions, can be obtained by looking at the problem from a Bayesian point of view. A result which follows from a general Bayes solution is that the inverse estimator [4] is Bayes with respect to a particular informative prior.
TL;DR: In the household survey with multiplicity, sample households report information about their own residents as well as about other persons who live elsewhere, such as relatives or neighbors, as specified by a multiplicity rule adopted in the survey as discussed by the authors.
Abstract: In the household survey with multiplicity, sample households report information about their own residents as well as about other persons who live elsewhere, such as relatives or neighbors, as specified by a multiplicity rule adopted in the survey. Although sampling errors for the multiplicity survey are not necessarily smaller than those for the conventional survey in which sample households report for their own residents only, in most instances it should be feasible to assure a substantial reduction in sampling error by selecting appropriate multiplicity rules. Using alternative statistical models, it is demonstrated that under specified conditions, sampling errors for the multiplicity survey are necessarily smaller than those for the conventional survey, and the results give insight regarding the factors contributing to the efficiency of the multiplicity survey.
TL;DR: In this article, the null distribution of the chi-square goodness of fit statistic, X 2, is examined numerically for many different multinomial distributions and it is proven that the C(m) distribution is the limiting distribution of X 2 when some expectations remain finite while the rest increase without limit.
Abstract: The accuracy of approximations for the null distribution of the chi-square goodness of fit statistic, X 2, is examined numerically for many different multinomial distributions. Some of these approximations are terms of a new asymptotic expansion which provides much greater accuracy than the usual χ2 approximation. When there are too many small expectations, even these are inaccurate, and a diflerent type of approximation must be used—the C(m) distribution. It is proven that the latter is the limiting distribution of X 2 when some expectations remain finite while the rest increase without limit, and the limiting multivariate distribution of the multinomial under these conditions is also derived. The C(m) approximation is used to study the error in the χ2 approximation when some expectations are small, resulting in a new rule for the use of the upper one and five percent points of the χ2 approximation. This rule allows the use of arbitrarily small expectations, e.g., 1. Similar results are obtained...
TL;DR: In this paper, a linear least square test based on fifth degree polynomials has been run on more than twenty different computer programs in order to assess their numerical accuracy, and it was found that those programs using orthogonal Householder transformations, classical Gram-Schmidt orthonormalization or modified GramSchmidt Orthogonalization were generally much more accurate than those using elimination algorithms.
Abstract: Linear least squares test problems based on fifth degree polynomials have been run on more than twenty different computer programs in order to assess their numerical accuracy. The programs tested, all in present-day use, included representatives from several statistical packages as well as some from the SHARE library. Essentially five different algorithms were used in the various programs to obtain the coefficients of the least squares fits. The tests were run on several different computers, in double precision as well as single precision. By comparing the coefficients reported, it was found that those programs using orthogonal Householder transformations, classical Gram-Schmidt orthonormalization or modified Gram-Schmidt orthogonalization were generally much more accurate than those using elimination algorithms. Programs using orthogonal polynomials (suitable only for polynomial fits) also proved to be superior to those using elimination algorithms. The most successful programs accumulated inner...
TL;DR: In this article, the maximum likelihood estimators of the parameters of the generalized gamma distribution are shown to have the property that are distributed independently of a and b. Similar properties are also obtained for some Weibull-type statistics.
Abstract: The maximum likelihood estimators of the parameters of the generalized gamma distribution are shown to have the property that are distributed independently of a and b. Similar properties are also obtained for some Weibull-type statistics. These results are applied in particular to the problem of discriminating between the Weibull model and the generalized gamma model, and several test statistics are considered. In general, the Weibull model is quite flexible and, unless the sample size is quite large, is perhaps a preferable assumption because of the computational complexity and other difficulties encountered with the generalized gamma distribution.